# Properties

 Label 1134.2.f.m Level $1134$ Weight $2$ Character orbit 1134.f Analytic conductor $9.055$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} - q^{8} + q^{10} + ( 2 - 2 \zeta_{6} ) q^{11} -3 \zeta_{6} q^{13} -\zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + q^{17} + 2 q^{19} + ( 1 - \zeta_{6} ) q^{20} -2 \zeta_{6} q^{22} + 2 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} -3 q^{26} - q^{28} + ( 7 - 7 \zeta_{6} ) q^{29} -6 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 1 - \zeta_{6} ) q^{34} + q^{35} -7 q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} -\zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -2 q^{44} + 2 q^{46} + ( 6 - 6 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} -4 \zeta_{6} q^{50} + ( -3 + 3 \zeta_{6} ) q^{52} + 6 q^{53} + 2 q^{55} + ( -1 + \zeta_{6} ) q^{56} -7 \zeta_{6} q^{58} + 10 \zeta_{6} q^{59} + ( 9 - 9 \zeta_{6} ) q^{61} -6 q^{62} + q^{64} + ( 3 - 3 \zeta_{6} ) q^{65} -10 \zeta_{6} q^{67} -\zeta_{6} q^{68} + ( 1 - \zeta_{6} ) q^{70} + 4 q^{71} -11 q^{73} + ( -7 + 7 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{76} -2 \zeta_{6} q^{77} + ( 6 - 6 \zeta_{6} ) q^{79} - q^{80} + 6 q^{82} + ( -10 + 10 \zeta_{6} ) q^{83} + \zeta_{6} q^{85} -4 \zeta_{6} q^{86} + ( -2 + 2 \zeta_{6} ) q^{88} -15 q^{89} -3 q^{91} + ( 2 - 2 \zeta_{6} ) q^{92} -6 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + ( 2 - 2 \zeta_{6} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + q^{5} + q^{7} - 2q^{8} + 2q^{10} + 2q^{11} - 3q^{13} - q^{14} - q^{16} + 2q^{17} + 4q^{19} + q^{20} - 2q^{22} + 2q^{23} + 4q^{25} - 6q^{26} - 2q^{28} + 7q^{29} - 6q^{31} + q^{32} + q^{34} + 2q^{35} - 14q^{37} + 2q^{38} - q^{40} + 6q^{41} + 4q^{43} - 4q^{44} + 4q^{46} + 6q^{47} - q^{49} - 4q^{50} - 3q^{52} + 12q^{53} + 4q^{55} - q^{56} - 7q^{58} + 10q^{59} + 9q^{61} - 12q^{62} + 2q^{64} + 3q^{65} - 10q^{67} - q^{68} + q^{70} + 8q^{71} - 22q^{73} - 7q^{74} - 2q^{76} - 2q^{77} + 6q^{79} - 2q^{80} + 12q^{82} - 10q^{83} + q^{85} - 4q^{86} - 2q^{88} - 30q^{89} - 6q^{91} + 2q^{92} - 6q^{94} + 2q^{95} + 2q^{97} - 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 0 1.00000
757.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 0 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.f.m 2
3.b odd 2 1 1134.2.f.d 2
9.c even 3 1 1134.2.a.b 1
9.c even 3 1 inner 1134.2.f.m 2
9.d odd 6 1 1134.2.a.g yes 1
9.d odd 6 1 1134.2.f.d 2
36.f odd 6 1 9072.2.a.k 1
36.h even 6 1 9072.2.a.p 1
63.l odd 6 1 7938.2.a.j 1
63.o even 6 1 7938.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.b 1 9.c even 3 1
1134.2.a.g yes 1 9.d odd 6 1
1134.2.f.d 2 3.b odd 2 1
1134.2.f.d 2 9.d odd 6 1
1134.2.f.m 2 1.a even 1 1 trivial
1134.2.f.m 2 9.c even 3 1 inner
7938.2.a.j 1 63.l odd 6 1
7938.2.a.w 1 63.o even 6 1
9072.2.a.k 1 36.f odd 6 1
9072.2.a.p 1 36.h even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1134, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13}^{2} + 3 T_{13} + 9$$ $$T_{17} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$9 + 3 T + T^{2}$$
$17$ $$( -1 + T )^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$4 - 2 T + T^{2}$$
$29$ $$49 - 7 T + T^{2}$$
$31$ $$36 + 6 T + T^{2}$$
$37$ $$( 7 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$100 - 10 T + T^{2}$$
$61$ $$81 - 9 T + T^{2}$$
$67$ $$100 + 10 T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$( 11 + T )^{2}$$
$79$ $$36 - 6 T + T^{2}$$
$83$ $$100 + 10 T + T^{2}$$
$89$ $$( 15 + T )^{2}$$
$97$ $$4 - 2 T + T^{2}$$